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<a href="example3_8h.html">Go to the documentation of this file.</a><div class="fragment"><div class="line"><a name="l00001"></a><span class="lineno">    1</span>&#160;<span class="comment">/**</span></div>
<div class="line"><a name="l00002"></a><span class="lineno">    2</span>&#160;<span class="comment"> * @page example3 Example 3 : Cahn-Hilliard (two species)</span></div>
<div class="line"><a name="l00003"></a><span class="lineno">    3</span>&#160;<span class="comment"> * \dontinclude CahnHilliard_twoSpecies/2D/userFunctions.cc</span></div>
<div class="line"><a name="l00004"></a><span class="lineno">    4</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00005"></a><span class="lineno">    5</span>&#160;<span class="comment"> * This example implements the Cahn-Hilliard equation for phase-field modeling of a two species,</span></div>
<div class="line"><a name="l00006"></a><span class="lineno">    6</span>&#160;<span class="comment"> * as described by the following weak form of the PDE. The scalar fields are composition one, \f$c_1\f$, </span></div>
<div class="line"><a name="l00007"></a><span class="lineno">    7</span>&#160;<span class="comment"> * and composition two, \f$c_2\f$.</span></div>
<div class="line"><a name="l00008"></a><span class="lineno">    8</span>&#160;<span class="comment"> * Note the application of the higher-order Dirichlet boundary conditions \f$\nabla c_i\cdot\boldsymbol{n}=0\f$</span></div>
<div class="line"><a name="l00009"></a><span class="lineno">    9</span>&#160;<span class="comment"> * using Nitsche&#39;s method.</span></div>
<div class="line"><a name="l00010"></a><span class="lineno">   10</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00011"></a><span class="lineno">   11</span>&#160;<span class="comment"> * Cahn-Hilliard:</span></div>
<div class="line"><a name="l00012"></a><span class="lineno">   12</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00013"></a><span class="lineno">   13</span>&#160;<span class="comment"> * \f{eqnarray*}{</span></div>
<div class="line"><a name="l00014"></a><span class="lineno">   14</span>&#160;<span class="comment"> * 0 &amp;=&amp; \int_\Omega \left(w_1\frac{c_1 - c_{1_{prev}}}{\mathrm{d}t} + </span></div>
<div class="line"><a name="l00015"></a><span class="lineno">   15</span>&#160;<span class="comment"> * M\left(\nabla w_1\cdot(f_{,c_1c_1}\nabla c_1) + f_{,c_1c_2}\nabla c_2) + \kappa_1\nabla^2 w_1\nabla^2 c_1\right)\right) dV\\</span></div>
<div class="line"><a name="l00016"></a><span class="lineno">   16</span>&#160;<span class="comment"> * &amp;\phantom{=}&amp; - \int_{\partial\Omega} \left(w_1j_n + </span></div>
<div class="line"><a name="l00017"></a><span class="lineno">   17</span>&#160;<span class="comment"> * M\kappa_1\left(\nabla^2c_1(\nabla w_1\cdot\boldsymbol{n}) + \nabla^2w_1(\nabla c_1\cdot\boldsymbol{n})\right)</span></div>
<div class="line"><a name="l00018"></a><span class="lineno">   18</span>&#160;<span class="comment"> *  - \tau(\nabla w_1\cdot\boldsymbol{n})(\nabla c_1\cdot\boldsymbol{n})\right) dS\\</span></div>
<div class="line"><a name="l00019"></a><span class="lineno">   19</span>&#160;<span class="comment"> * 0 &amp;=&amp; \int_\Omega \left(w_2\frac{c_2 - c_{2_{prev}}}{\mathrm{d}t} + </span></div>
<div class="line"><a name="l00020"></a><span class="lineno">   20</span>&#160;<span class="comment"> * M\left(\nabla w_2\cdot(f_{,c_2c_1}\nabla c_1) + f_{,c_2c_2}\nabla c_2) + \kappa_1\nabla^2 w_2\nabla^2 c_2\right)\right) dV\\</span></div>
<div class="line"><a name="l00021"></a><span class="lineno">   21</span>&#160;<span class="comment"> * &amp;\phantom{=}&amp; - \int_{\partial\Omega} \left(w_2j_n + </span></div>
<div class="line"><a name="l00022"></a><span class="lineno">   22</span>&#160;<span class="comment"> * M\kappa_1\left(\nabla^2c_2(\nabla w_2\cdot\boldsymbol{n}) + \nabla^2w_2(\nabla c_2\cdot\boldsymbol{n})\right)</span></div>
<div class="line"><a name="l00023"></a><span class="lineno">   23</span>&#160;<span class="comment"> *  - \tau(\nabla w_2\cdot\boldsymbol{n})(\nabla c_2\cdot\boldsymbol{n})\right) dS</span></div>
<div class="line"><a name="l00024"></a><span class="lineno">   24</span>&#160;<span class="comment"> * \f}</span></div>
<div class="line"><a name="l00025"></a><span class="lineno">   25</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00026"></a><span class="lineno">   26</span>&#160;<span class="comment"> * Free energy density:</span></div>
<div class="line"><a name="l00027"></a><span class="lineno">   27</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00028"></a><span class="lineno">   28</span>&#160;<span class="comment"> * \f{eqnarray*}{</span></div>
<div class="line"><a name="l00029"></a><span class="lineno">   29</span>&#160;<span class="comment"> * f(c_1,c_2) = c_2(c_1-0.1)^2(c_1-0.9)^2 + (c_2-0.2)^2(c_2-0.8)^2</span></div>
<div class="line"><a name="l00030"></a><span class="lineno">   30</span>&#160;<span class="comment"> * \f}</span></div>
<div class="line"><a name="l00031"></a><span class="lineno">   31</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00032"></a><span class="lineno">   32</span>&#160;<span class="comment"> * The current settings prescribe random initial conditions and zero-flux</span></div>
<div class="line"><a name="l00033"></a><span class="lineno">   33</span>&#160;<span class="comment"> * boundary conditions. With these settings, the following evolution </span></div>
<div class="line"><a name="l00034"></a><span class="lineno">   34</span>&#160;<span class="comment"> * of the concentration of the two species is obtained:</span></div>
<div class="line"><a name="l00035"></a><span class="lineno">   35</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00036"></a><span class="lineno">   36</span>&#160;<span class="comment"> * \htmlonly &lt;style&gt;div.image img[src=&quot;example3.png&quot;]{width:20cm;}&lt;/style&gt; \endhtmlonly</span></div>
<div class="line"><a name="l00037"></a><span class="lineno">   37</span>&#160;<span class="comment"> * @image html example3.png </span></div>
<div class="line"><a name="l00038"></a><span class="lineno">   38</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00039"></a><span class="lineno">   39</span>&#160;<span class="comment"> * Implementation: Level 1 users</span></div>
<div class="line"><a name="l00040"></a><span class="lineno">   40</span>&#160;<span class="comment"> * ==============================</span></div>
<div class="line"><a name="l00041"></a><span class="lineno">   41</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00042"></a><span class="lineno">   42</span>&#160;<span class="comment"> * To implement this model, we will specify the following through defining user functions: &lt;br&gt;</span></div>
<div class="line"><a name="l00043"></a><span class="lineno">   43</span>&#160;<span class="comment"> * - Initial conditions &lt;br&gt;</span></div>
<div class="line"><a name="l00044"></a><span class="lineno">   44</span>&#160;<span class="comment"> * - Constitutive model (via free energy density functions) &lt;br&gt;</span></div>
<div class="line"><a name="l00045"></a><span class="lineno">   45</span>&#160;<span class="comment"> * - Parameter values &lt;br&gt;</span></div>
<div class="line"><a name="l00046"></a><span class="lineno">   46</span>&#160;<span class="comment"> * - Weak form of the PDEs &lt;br&gt;</span></div>
<div class="line"><a name="l00047"></a><span class="lineno">   47</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00048"></a><span class="lineno">   48</span>&#160;<span class="comment"> * First, we include the header file declaring the required user functions. These functions will be defined in this file.</span></div>
<div class="line"><a name="l00049"></a><span class="lineno">   49</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00050"></a><span class="lineno">   50</span>&#160;<span class="comment"> * \line userFunctions</span></div>
<div class="line"><a name="l00051"></a><span class="lineno">   51</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00052"></a><span class="lineno">   52</span>&#160;<span class="comment"> * Now, we first define any optional user functions. Optional user functions have a default definition that can be </span></div>
<div class="line"><a name="l00053"></a><span class="lineno">   53</span>&#160;<span class="comment"> * redefined by the user using a function pointer.</span></div>
<div class="line"><a name="l00054"></a><span class="lineno">   54</span>&#160;<span class="comment"> * This will be done in the \c defineParameters function. The available list of optional user functions includes:</span></div>
<div class="line"><a name="l00055"></a><span class="lineno">   55</span>&#160;<span class="comment"> * \c boundaryConditions, \c scalarInitialConditions, \c vectorInitialConditions, \c loadStep, \c adaptiveTimeStep, and \c projectFields.</span></div>
<div class="line"><a name="l00056"></a><span class="lineno">   56</span>&#160;<span class="comment"> * In this example, we redefine only the \c scalarInitialConditions function, while using the default functions for the others.</span></div>
<div class="line"><a name="l00057"></a><span class="lineno">   57</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00058"></a><span class="lineno">   58</span>&#160;<span class="comment"> * &lt;b&gt; The \c scalarInitialConditions function &lt;/b&gt;</span></div>
<div class="line"><a name="l00059"></a><span class="lineno">   59</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00060"></a><span class="lineno">   60</span>&#160;<span class="comment"> * We initialized the two composition fields to be random about 0.5.</span></div>
<div class="line"><a name="l00061"></a><span class="lineno">   61</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00062"></a><span class="lineno">   62</span>&#160;<span class="comment"> * \skip template</span></div>
<div class="line"><a name="l00063"></a><span class="lineno">   63</span>&#160;<span class="comment"> * \until //end</span></div>
<div class="line"><a name="l00064"></a><span class="lineno">   64</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00065"></a><span class="lineno">   65</span>&#160;<span class="comment"> * &lt;b&gt; Free energy density derivative functions &lt;/b&gt;</span></div>
<div class="line"><a name="l00066"></a><span class="lineno">   66</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00067"></a><span class="lineno">   67</span>&#160;<span class="comment"> * This phase-field implementation requires the first and second derivatives of the chemical free energy density function</span></div>
<div class="line"><a name="l00068"></a><span class="lineno">   68</span>&#160;<span class="comment"> * \f$f(c_1,c_2) = c_2(c_1-0.1)^2(c_1-0.9)^2 + (c_2-0.2)^2(c_2-0.8)^2\f$. We define the function computing</span></div>
<div class="line"><a name="l00069"></a><span class="lineno">   69</span>&#160;<span class="comment"> * \f$\partial^2 f/\partial c \partial c\f$ here.</span></div>
<div class="line"><a name="l00070"></a><span class="lineno">   70</span>&#160;<span class="comment"> * Note that this free energy derivative function is used only in this file. It is not a member of any class,</span></div>
<div class="line"><a name="l00071"></a><span class="lineno">   71</span>&#160;<span class="comment"> * nor will we use it to set any function pointers.</span></div>
<div class="line"><a name="l00072"></a><span class="lineno">   72</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00073"></a><span class="lineno">   73</span>&#160;<span class="comment"> * \skip template</span></div>
<div class="line"><a name="l00074"></a><span class="lineno">   74</span>&#160;<span class="comment"> * \until //end</span></div>
<div class="line"><a name="l00075"></a><span class="lineno">   75</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00076"></a><span class="lineno">   76</span>&#160;<span class="comment"> * &lt;b&gt; The \c defineParameters function &lt;/b&gt;</span></div>
<div class="line"><a name="l00077"></a><span class="lineno">   77</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00078"></a><span class="lineno">   78</span>&#160;<span class="comment"> * The user is required to define the \c defineParameters and \c residual functions. The \c defineParameters defines variables</span></div>
<div class="line"><a name="l00079"></a><span class="lineno">   79</span>&#160;<span class="comment"> * and functions in the \c AppCtx object. The \c AppCtx object is defined</span></div>
<div class="line"><a name="l00080"></a><span class="lineno">   80</span>&#160;<span class="comment"> * in the appCtx.h file. This function is used to define any values in \c user that will be needed in the problem.</span></div>
<div class="line"><a name="l00081"></a><span class="lineno">   81</span>&#160;<span class="comment"> * It is also used to set any function pointers for user functions that we have redefined.</span></div>
<div class="line"><a name="l00082"></a><span class="lineno">   82</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00083"></a><span class="lineno">   83</span>&#160;<span class="comment"> * Many of these values can be overwritten by the parameters.prm file, which we will look at later.</span></div>
<div class="line"><a name="l00084"></a><span class="lineno">   84</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00085"></a><span class="lineno">   85</span>&#160;<span class="comment"> * \skip template</span></div>
<div class="line"><a name="l00086"></a><span class="lineno">   86</span>&#160;<span class="comment"> * \until void</span></div>
<div class="line"><a name="l00087"></a><span class="lineno">   87</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00088"></a><span class="lineno">   88</span>&#160;<span class="comment"> * Here, we define the mesh by setting the number of elements in each direction, e.g. a 100x100 element mesh.</span></div>
<div class="line"><a name="l00089"></a><span class="lineno">   89</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00090"></a><span class="lineno">   90</span>&#160;<span class="comment"> * \skip user.N[0]</span></div>
<div class="line"><a name="l00091"></a><span class="lineno">   91</span>&#160;<span class="comment"> * \until user.N[1]</span></div>
<div class="line"><a name="l00092"></a><span class="lineno">   92</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00093"></a><span class="lineno">   93</span>&#160;<span class="comment"> * We also define the dimensions of the domain, e.g. a unit square.</span></div>
<div class="line"><a name="l00094"></a><span class="lineno">   94</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00095"></a><span class="lineno">   95</span>&#160;<span class="comment"> * \skip user.L[0]</span></div>
<div class="line"><a name="l00096"></a><span class="lineno">   96</span>&#160;<span class="comment"> * \until user.L[1]</span></div>
<div class="line"><a name="l00097"></a><span class="lineno">   97</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00098"></a><span class="lineno">   98</span>&#160;<span class="comment"> * We can define additional material parameters that are not explicity listed in the \c user structure by</span></div>
<div class="line"><a name="l00099"></a><span class="lineno">   99</span>&#160;<span class="comment"> * defining elements of the \c matParam C++ map, which maps \c std::string to \c double. These values can also be overwritten</span></div>
<div class="line"><a name="l00100"></a><span class="lineno">  100</span>&#160;<span class="comment"> * in the parameters file.</span></div>
<div class="line"><a name="l00101"></a><span class="lineno">  101</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00102"></a><span class="lineno">  102</span>&#160;<span class="comment"> * \skip &quot;mobility&quot;</span></div>
<div class="line"><a name="l00103"></a><span class="lineno">  103</span>&#160;<span class="comment"> * \until &quot;kappa&quot;</span></div>
<div class="line"><a name="l00104"></a><span class="lineno">  104</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00105"></a><span class="lineno">  105</span>&#160;<span class="comment"> * We define the initial time step and total simulation time. We also have the options to use restart files, in which case</span></div>
<div class="line"><a name="l00106"></a><span class="lineno">  106</span>&#160;<span class="comment"> * we would set the iteration index and time at which to start. We leave these values at zero to begin a new simulation.</span></div>
<div class="line"><a name="l00107"></a><span class="lineno">  107</span>&#160;<span class="comment"> * We also have the option to output results at regular intervals (e.g. every 5 time steps).</span></div>
<div class="line"><a name="l00108"></a><span class="lineno">  108</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00109"></a><span class="lineno">  109</span>&#160;<span class="comment"> * \skip user.dtVal</span></div>
<div class="line"><a name="l00110"></a><span class="lineno">  110</span>&#160;<span class="comment"> * \until user.skipOutput</span></div>
<div class="line"><a name="l00111"></a><span class="lineno">  111</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00112"></a><span class="lineno">  112</span>&#160;<span class="comment"> * We specify the number of vector and scalar solution and projection fields by adding the name of each field to</span></div>
<div class="line"><a name="l00113"></a><span class="lineno">  113</span>&#160;<span class="comment"> * their respective vector. Here, we have one scalar solution field (the composition).</span></div>
<div class="line"><a name="l00114"></a><span class="lineno">  114</span>&#160;<span class="comment"> * We do not use any vector solution fields or projection fields in this example.</span></div>
<div class="line"><a name="l00115"></a><span class="lineno">  115</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00116"></a><span class="lineno">  116</span>&#160;<span class="comment"> * \skip &quot;c1&quot;</span></div>
<div class="line"><a name="l00117"></a><span class="lineno">  117</span>&#160;<span class="comment"> * \until &quot;c2&quot;</span></div>
<div class="line"><a name="l00118"></a><span class="lineno">  118</span>&#160;<span class="comment"> * </span></div>
<div class="line"><a name="l00119"></a><span class="lineno">  119</span>&#160;<span class="comment"> * We can specify the polynomial order of the basis splines, as well as the global continuity.</span></div>
<div class="line"><a name="l00120"></a><span class="lineno">  120</span>&#160;<span class="comment"> * Note that the global continuity must be less than the polynomial order.</span></div>
<div class="line"><a name="l00121"></a><span class="lineno">  121</span>&#160;<span class="comment"> * Here, we use quadratic basis functions with C-1 global continuity.</span></div>
<div class="line"><a name="l00122"></a><span class="lineno">  122</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00123"></a><span class="lineno">  123</span>&#160;<span class="comment"> * \skip polyOrder</span></div>
<div class="line"><a name="l00124"></a><span class="lineno">  124</span>&#160;<span class="comment"> * \until globalContinuity</span></div>
<div class="line"><a name="l00125"></a><span class="lineno">  125</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00126"></a><span class="lineno">  126</span>&#160;<span class="comment"> * Finally, we redirect the desired user function pointers to the \c scalarInitialConditions function that we</span></div>
<div class="line"><a name="l00127"></a><span class="lineno">  127</span>&#160;<span class="comment"> * defined above. This completes the \c defineParameters function.</span></div>
<div class="line"><a name="l00128"></a><span class="lineno">  128</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00129"></a><span class="lineno">  129</span>&#160;<span class="comment"> * \skip scalarInitialConditions</span></div>
<div class="line"><a name="l00130"></a><span class="lineno">  130</span>&#160;<span class="comment"> * \until //end</span></div>
<div class="line"><a name="l00131"></a><span class="lineno">  131</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00132"></a><span class="lineno">  132</span>&#160;<span class="comment"> * &lt;b&gt; The \c residual function &lt;/b&gt;</span></div>
<div class="line"><a name="l00133"></a><span class="lineno">  133</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00134"></a><span class="lineno">  134</span>&#160;<span class="comment"> * The residual function defines the residual that is to be driven to zero.</span></div>
<div class="line"><a name="l00135"></a><span class="lineno">  135</span>&#160;<span class="comment"> * This is the central function of the code.</span></div>
<div class="line"><a name="l00136"></a><span class="lineno">  136</span>&#160;<span class="comment"> * It is set up to follow the analytical weak form of the PDE.</span></div>
<div class="line"><a name="l00137"></a><span class="lineno">  137</span>&#160;<span class="comment"> * It has a number of arguments that give problem information at the current quadrature point.</span></div>
<div class="line"><a name="l00138"></a><span class="lineno">  138</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00139"></a><span class="lineno">  139</span>&#160;<span class="comment"> * \skip template</span></div>
<div class="line"><a name="l00140"></a><span class="lineno">  140</span>&#160;<span class="comment"> * \until &amp;r</span></div>
<div class="line"><a name="l00141"></a><span class="lineno">  141</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00142"></a><span class="lineno">  142</span>&#160;<span class="comment"> * \c dV is a boolean, &quot;true&quot; if \c residual is being called for the volume integral and </span></div>
<div class="line"><a name="l00143"></a><span class="lineno">  143</span>&#160;<span class="comment"> * &quot;false&quot; if \c residual is being called for the surface integral.\n</span></div>
<div class="line"><a name="l00144"></a><span class="lineno">  144</span>&#160;<span class="comment"> * \c dS is a boolean, &quot;false&quot; if \c residual is being called for the volume integral and </span></div>
<div class="line"><a name="l00145"></a><span class="lineno">  145</span>&#160;<span class="comment"> * &quot;true&quot; if \c residual is being called for the surface integral.\n</span></div>
<div class="line"><a name="l00146"></a><span class="lineno">  146</span>&#160;<span class="comment"> * \c x gives the coordinates of the quadrature point.\n</span></div>
<div class="line"><a name="l00147"></a><span class="lineno">  147</span>&#160;<span class="comment"> * \c normal gives the unit normal for a surface quadrature point.\n</span></div>
<div class="line"><a name="l00148"></a><span class="lineno">  148</span>&#160;<span class="comment"> * \c c gives the information (values, gradients, etc.) for the scalar solution fields at the current quadrature point</span></div>
<div class="line"><a name="l00149"></a><span class="lineno">  149</span>&#160;<span class="comment"> * (see documentation for solutionScalars class).\n</span></div>
<div class="line"><a name="l00150"></a><span class="lineno">  150</span>&#160;<span class="comment"> * \c u gives the information (values, gradients, etc.) for the vector solution fields at the current quadrature point</span></div>
<div class="line"><a name="l00151"></a><span class="lineno">  151</span>&#160;<span class="comment"> * (see documentation for solutionVectors class).\n</span></div>
<div class="line"><a name="l00152"></a><span class="lineno">  152</span>&#160;<span class="comment"> * \c w1 gives the information for the scalar test functions.\n</span></div>
<div class="line"><a name="l00153"></a><span class="lineno">  153</span>&#160;<span class="comment"> * \c w2 gives the information for the vector test functions.\n</span></div>
<div class="line"><a name="l00154"></a><span class="lineno">  154</span>&#160;<span class="comment"> * \c user is a structure available for parameters related to the initial boundary value problem (e.g. elasticity tensor).\n</span></div>
<div class="line"><a name="l00155"></a><span class="lineno">  155</span>&#160;<span class="comment"> * \c r stores the scalar value of the residual for the weak form of the PDE which is then used by the core assembly functions.</span></div>
<div class="line"><a name="l00156"></a><span class="lineno">  156</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00157"></a><span class="lineno">  157</span>&#160;<span class="comment"> * The following functions are available for the solution objects \c c and \c u,</span></div>
<div class="line"><a name="l00158"></a><span class="lineno">  158</span>&#160;<span class="comment"> * where the argument is the field index, i.</span></div>
<div class="line"><a name="l00159"></a><span class="lineno">  159</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00160"></a><span class="lineno">  160</span>&#160;<span class="comment"> * \c c.val(i) - Value of scalar field i, scalar \n</span></div>
<div class="line"><a name="l00161"></a><span class="lineno">  161</span>&#160;<span class="comment"> * \c c.grad(i) - Gradient of scalar field i, 1st order tensor \n</span></div>
<div class="line"><a name="l00162"></a><span class="lineno">  162</span>&#160;<span class="comment"> * \c c.hess(i) - Hessian of scalar field i, 2nd order tensor \n</span></div>
<div class="line"><a name="l00163"></a><span class="lineno">  163</span>&#160;<span class="comment"> * \c c.laplacian(i) - Laplacian of scalar field i, scalar \n</span></div>
<div class="line"><a name="l00164"></a><span class="lineno">  164</span>&#160;<span class="comment"> * \c c.valP(i) - Value of scalar field i at previous time step, scalar \n</span></div>
<div class="line"><a name="l00165"></a><span class="lineno">  165</span>&#160;<span class="comment"> * \c c.gradP(i) - Gradient of scalar field i at previous time step, 1st order tensor \n</span></div>
<div class="line"><a name="l00166"></a><span class="lineno">  166</span>&#160;<span class="comment"> * \c c.hessP(i) - Hessian of scalar field i at previous time step, 2nd order tensor \n</span></div>
<div class="line"><a name="l00167"></a><span class="lineno">  167</span>&#160;<span class="comment"> * \c c.laplacianP(i) - Laplacian of scalar field i at previous time step, scalar</span></div>
<div class="line"><a name="l00168"></a><span class="lineno">  168</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00169"></a><span class="lineno">  169</span>&#160;<span class="comment"> * \c u.val(i) - Value of vector field i, 1st order tensor \n</span></div>
<div class="line"><a name="l00170"></a><span class="lineno">  170</span>&#160;<span class="comment"> * \c u.grad(i) - Gradient of vector field i, 2nd order tensor \n</span></div>
<div class="line"><a name="l00171"></a><span class="lineno">  171</span>&#160;<span class="comment"> * \c u.hess(i) - Hessian of vector field i, 3rd order tensor \n</span></div>
<div class="line"><a name="l00172"></a><span class="lineno">  172</span>&#160;<span class="comment"> * \c u.valP(i) - Value of vector field i at previous time step, 1st order tensor \n</span></div>
<div class="line"><a name="l00173"></a><span class="lineno">  173</span>&#160;<span class="comment"> * \c u.gradP(i) - Gradient of vector field i at previous time step, 2nd order tensor \n</span></div>
<div class="line"><a name="l00174"></a><span class="lineno">  174</span>&#160;<span class="comment"> * \c u.hessP(i) - Hessian of vector field i at previous time step, 3rd order tensor</span></div>
<div class="line"><a name="l00175"></a><span class="lineno">  175</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00176"></a><span class="lineno">  176</span>&#160;<span class="comment"> * Similar functions are available for the test functions. Also, the following tensor operations are useful:</span></div>
<div class="line"><a name="l00177"></a><span class="lineno">  177</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00178"></a><span class="lineno">  178</span>&#160;<span class="comment"> * Tensor operations: \n</span></div>
<div class="line"><a name="l00179"></a><span class="lineno">  179</span>&#160;<span class="comment"> * \c operator+ - tensor addition \n</span></div>
<div class="line"><a name="l00180"></a><span class="lineno">  180</span>&#160;<span class="comment"> * \c operator- - tensor subraction \n</span></div>
<div class="line"><a name="l00181"></a><span class="lineno">  181</span>&#160;<span class="comment"> * \c operator* - single contraction between tensors or scalar multiplication \n</span></div>
<div class="line"><a name="l00182"></a><span class="lineno">  182</span>&#160;<span class="comment"> * \c double_contract - double contraction of two 2nd order tensors, or</span></div>
<div class="line"><a name="l00183"></a><span class="lineno">  183</span>&#160;<span class="comment"> *                   a 4th order tensor and a 2nd order tensor. \n</span></div>
<div class="line"><a name="l00184"></a><span class="lineno">  184</span>&#160;<span class="comment"> * \c trans( ) - transpose 2nd order tensor \n</span></div>
<div class="line"><a name="l00185"></a><span class="lineno">  185</span>&#160;<span class="comment"> * \c trace( ) - trace of 2nd order tensor \n</span></div>
<div class="line"><a name="l00186"></a><span class="lineno">  186</span>&#160;<span class="comment"> * \c det( ) - determinant of 2nd order tensor \n</span></div>
<div class="line"><a name="l00187"></a><span class="lineno">  187</span>&#160;<span class="comment"> * \c inv( ) - inverse of 2nd order tensor \n</span></div>
<div class="line"><a name="l00188"></a><span class="lineno">  188</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00189"></a><span class="lineno">  189</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00190"></a><span class="lineno">  190</span>&#160;<span class="comment"> * The example code here implements the weak form for the Cahn-Hilliard equations, as shown above.</span></div>
<div class="line"><a name="l00191"></a><span class="lineno">  191</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00192"></a><span class="lineno">  192</span>&#160;<span class="comment"> * First, we set the values for necessary parameters, using some predefined material parameters.</span></div>
<div class="line"><a name="l00193"></a><span class="lineno">  193</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00194"></a><span class="lineno">  194</span>&#160;<span class="comment"> * \skip dt</span></div>
<div class="line"><a name="l00195"></a><span class="lineno">  195</span>&#160;<span class="comment"> * \until tau</span></div>
<div class="line"><a name="l00196"></a><span class="lineno">  196</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00197"></a><span class="lineno">  197</span>&#160;<span class="comment"> * Next, we get the values for the free energy derivatives based on the current quadrature point.</span></div>
<div class="line"><a name="l00198"></a><span class="lineno">  198</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00199"></a><span class="lineno">  199</span>&#160;<span class="comment"> * \skip f_c1c1,</span></div>
<div class="line"><a name="l00200"></a><span class="lineno">  200</span>&#160;<span class="comment"> * \until F_c2c2(c.val(0),c.val(1));</span></div>
<div class="line"><a name="l00201"></a><span class="lineno">  201</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00202"></a><span class="lineno">  202</span>&#160;<span class="comment"> * Now, we compute the residual in a manner very similar to the analytical form:</span></div>
<div class="line"><a name="l00203"></a><span class="lineno">  203</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00204"></a><span class="lineno">  204</span>&#160;<span class="comment"> * \f{eqnarray*}{</span></div>
<div class="line"><a name="l00205"></a><span class="lineno">  205</span>&#160;<span class="comment"> * 0 &amp;=&amp; \int_\Omega \left(w_1\frac{c_1 - c_{1_{prev}}}{\mathrm{d}t} + </span></div>
<div class="line"><a name="l00206"></a><span class="lineno">  206</span>&#160;<span class="comment"> * M\left(\nabla w_1\cdot(f_{,c_1c_1}\nabla c_1) + f_{,c_1c_2}\nabla c_2) + \kappa_1\nabla^2 w_1\nabla^2 c_1\right)\right) dV\\</span></div>
<div class="line"><a name="l00207"></a><span class="lineno">  207</span>&#160;<span class="comment"> * &amp;\phantom{=}&amp; - \int_{\partial\Omega} \left(w_1j_n + </span></div>
<div class="line"><a name="l00208"></a><span class="lineno">  208</span>&#160;<span class="comment"> * M\kappa_1\left(\nabla^2c_1(\nabla w_1\cdot\boldsymbol{n}) + \nabla^2w_1(\nabla c_1\cdot\boldsymbol{n})\right)</span></div>
<div class="line"><a name="l00209"></a><span class="lineno">  209</span>&#160;<span class="comment"> *  - \tau(\nabla w_1\cdot\boldsymbol{n})(\nabla c_1\cdot\boldsymbol{n})\right) dS\\</span></div>
<div class="line"><a name="l00210"></a><span class="lineno">  210</span>&#160;<span class="comment"> * 0 &amp;=&amp; \int_\Omega \left(w_2\frac{c_2 - c_{2_{prev}}}{\mathrm{d}t} + </span></div>
<div class="line"><a name="l00211"></a><span class="lineno">  211</span>&#160;<span class="comment"> * M\left(\nabla w_2\cdot(f_{,c_2c_1}\nabla c_1) + f_{,c_2c_2}\nabla c_2) + \kappa_1\nabla^2 w_2\nabla^2 c_2\right)\right) dV\\</span></div>
<div class="line"><a name="l00212"></a><span class="lineno">  212</span>&#160;<span class="comment"> * &amp;\phantom{=}&amp; - \int_{\partial\Omega} \left(w_2j_n + </span></div>
<div class="line"><a name="l00213"></a><span class="lineno">  213</span>&#160;<span class="comment"> * M\kappa_1\left(\nabla^2c_2(\nabla w_2\cdot\boldsymbol{n}) + \nabla^2w_2(\nabla c_2\cdot\boldsymbol{n})\right)</span></div>
<div class="line"><a name="l00214"></a><span class="lineno">  214</span>&#160;<span class="comment"> *  - \tau(\nabla w_2\cdot\boldsymbol{n})(\nabla c_2\cdot\boldsymbol{n})\right) dS</span></div>
<div class="line"><a name="l00215"></a><span class="lineno">  215</span>&#160;<span class="comment"> * \f}</span></div>
<div class="line"><a name="l00216"></a><span class="lineno">  216</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00217"></a><span class="lineno">  217</span>&#160;<span class="comment"> * \skip r =</span></div>
<div class="line"><a name="l00218"></a><span class="lineno">  218</span>&#160;<span class="comment"> * \until //end</span></div>
<div class="line"><a name="l00219"></a><span class="lineno">  219</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00220"></a><span class="lineno">  220</span>&#160;<span class="comment"> * Finally, we include a file that instatiates the template functions \c defineParameters and \c residual. This bit of code</span></div>
<div class="line"><a name="l00221"></a><span class="lineno">  221</span>&#160;<span class="comment"> * will generally be the same for any problem (unless you decide to use a different automatic differentation library);</span></div>
<div class="line"><a name="l00222"></a><span class="lineno">  222</span>&#160;<span class="comment"> * the user does not need to modify it.</span></div>
<div class="line"><a name="l00223"></a><span class="lineno">  223</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00224"></a><span class="lineno">  224</span>&#160;<span class="comment"> * \line userFunctionsInstantiation</span></div>
<div class="line"><a name="l00225"></a><span class="lineno">  225</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00226"></a><span class="lineno">  226</span>&#160;<span class="comment"> * The complete implementation can be found at  &lt;a href=&quot;https://github.com/mechanoChem/mechanoChemIGA/blob/master/initBounValProbs/CahnHilliard_twoSpecies/2D/userFunctions.cc&quot;&gt;Github&lt;/a&gt;.</span></div>
<div class="line"><a name="l00227"></a><span class="lineno">  227</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00228"></a><span class="lineno">  228</span>&#160;<span class="comment"> * Parameters file: Interface for level 2 users</span></div>
<div class="line"><a name="l00229"></a><span class="lineno">  229</span>&#160;<span class="comment"> * ==============================</span></div>
<div class="line"><a name="l00230"></a><span class="lineno">  230</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00231"></a><span class="lineno">  231</span>&#160;<span class="comment"> * Now let&#39;s look at the parameters file, \c parameters.prm. The advantages of the parameters file are that</span></div>
<div class="line"><a name="l00232"></a><span class="lineno">  232</span>&#160;<span class="comment"> * these values can be changed without recompiling the code and it can provide a clean interface to the code.</span></div>
<div class="line"><a name="l00233"></a><span class="lineno">  233</span>&#160;<span class="comment"> * \dontinclude CahnHilliard_twoSpecies/2D/parameters.prm</span></div>
<div class="line"><a name="l00234"></a><span class="lineno">  234</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00235"></a><span class="lineno">  235</span>&#160;<span class="comment"> * The parameters defined in the parameters file overwrite any previous values defined in the \c defineParameters function.</span></div>
<div class="line"><a name="l00236"></a><span class="lineno">  236</span>&#160;<span class="comment"> * Anything following the pound sign (#) is a comment. A parameter is defined using the syntax: </span></div>
<div class="line"><a name="l00237"></a><span class="lineno">  237</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00238"></a><span class="lineno">  238</span>&#160;<span class="comment"> * \c set \c parameterName \c = \c parameterValue</span></div>
<div class="line"><a name="l00239"></a><span class="lineno">  239</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00240"></a><span class="lineno">  240</span>&#160;<span class="comment"> * There is a set list of variables that can be read from the parameters file. Anything else will be added to </span></div>
<div class="line"><a name="l00241"></a><span class="lineno">  241</span>&#160;<span class="comment"> * the \c matParam structure with a double number type. Tensor objects can follow the format: 1 x 1 or [1,1] or (1,1), </span></div>
<div class="line"><a name="l00242"></a><span class="lineno">  242</span>&#160;<span class="comment"> * where the number of components must equal the spatial dimension of the problem.</span></div>
<div class="line"><a name="l00243"></a><span class="lineno">  243</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00244"></a><span class="lineno">  244</span>&#160;<span class="comment"> * In this example file, we begin by specifying the spatial dimension, the geometry dimensions, and the mesh size:</span></div>
<div class="line"><a name="l00245"></a><span class="lineno">  245</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00246"></a><span class="lineno">  246</span>&#160;<span class="comment"> * \skip dim</span></div>
<div class="line"><a name="l00247"></a><span class="lineno">  247</span>&#160;<span class="comment"> * \until set N</span></div>
<div class="line"><a name="l00248"></a><span class="lineno">  248</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00249"></a><span class="lineno">  249</span>&#160;<span class="comment"> * Next, we define some parameters that are specific to this problem,</span></div>
<div class="line"><a name="l00250"></a><span class="lineno">  250</span>&#160;<span class="comment"> * so they become elements of \c matParam (see the \c residual and \defineParameters functions above).</span></div>
<div class="line"><a name="l00251"></a><span class="lineno">  251</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00252"></a><span class="lineno">  252</span>&#160;<span class="comment"> * \skip mobility</span></div>
<div class="line"><a name="l00253"></a><span class="lineno">  253</span>&#160;<span class="comment"> * \until kappa</span></div>
<div class="line"><a name="l00254"></a><span class="lineno">  254</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00255"></a><span class="lineno">  255</span>&#160;<span class="comment"> * We then define time stepping, restart information, output frequency, and spline parameters.</span></div>
<div class="line"><a name="l00256"></a><span class="lineno">  256</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00257"></a><span class="lineno">  257</span>&#160;<span class="comment"> * \skip Time stepping</span></div>
<div class="line"><a name="l00258"></a><span class="lineno">  258</span>&#160;<span class="comment"> * \until globalContinuity</span></div>
<div class="line"><a name="l00259"></a><span class="lineno">  259</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00260"></a><span class="lineno">  260</span>&#160;<span class="comment"> * Note that we don&#39;t need to include all (or even any) of these parameters in this file. We defined default values previously.</span></div>
<div class="line"><a name="l00261"></a><span class="lineno">  261</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00262"></a><span class="lineno">  262</span>&#160;<span class="comment"> * The complete parameters file can be found at  &lt;a href=&quot;https://github.com/mechanoChem/mechanoChemIGA/blob/master/initBounValProbs/CahnHilliard_twoSpecies/2D/parameters.prm&quot;&gt;Github&lt;/a&gt;.</span></div>
<div class="line"><a name="l00263"></a><span class="lineno">  263</span>&#160;<span class="comment"> *</span></div>
<div class="line"><a name="l00264"></a><span class="lineno">  264</span>&#160;<span class="comment"> */</span></div>
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